A non-Golod ring with a trivial product on its Koszul homology

Abstract

We present a monomial ideal a⊂S such that S/a is not Golod, even though the product in its Koszul homology is trivial. This constitutes a counterexample to a well-known result by Berglund and Jöllenbeck (the error can be traced to a mistake in an earlier article by Jöllenbeck).On the positive side, we show that if R is a monomial ring such that the r-ary Massey product vanishes for all r≤max⁡(2,regR−2), then R is Golod. In particular, if R is the Stanley–Reisner ring of a simplicial complex of dimension at most 3, then R is Golod if and only if the product in its Koszul homology is trivial.Moreover, we show that if Δ is a triangulation of a k-orientable manifold whose Stanley–Reisner ring is Golod, then Δ is 2-neighborly. This extends a recent result of Iriye and Kishimoto.